Moon & Mars Orbiting Spinning Tether Transport - Tethers Unlimited

Tethers Unlimited, Inc.Appendixm M: TetherSim of the algorithms, the continuous tether mass is approximated as a series of point masses linked bymassless springs. In the explicit algorithm, the forces on each of the point masses are calculated andsummed, and Runge-Kutta integration is used to advance their positions over a timestep. Explicitpropagation of tether dynamics, however, requires the use of extremely small timesteps. Stability ofan explicit simulation scheme requires that the integration timestep be maintained smaller than thetime it takes for longitudinal and transverse waves to propagate along the length of a segment of thetether. The equations of tether oscillations with small deflections predict that these two modes willtravel with different velocities; the transverse velocity V t depends upon the tension T on the tether,and the longitudinal wave velocity V l depends upon the tether extensional stiffness E:TEV = V =tρwhere ρ is the linear density of the tether. To illustrate the challenge this poses, consider a very smalltether system that utilizes a 2 km long metal tether massing 1 kg. The linear density of the tether is 0.5g/m, the nominal gravity-gradient tension is approximately 0.2 N, and the tether stiffnessapproximately 5000 N/m. The transverse wave propagation speed will be roughly 20 m/s, and thelongitudinal wave propagation speed approximately 3162m/s. If the tether is modeled as twenty 100 msegments, the timestep must be lower than 0.03 s to maintain numerical stability. In practice, the truestability limit on the timestep is even smaller, as low as 0.003s, depending upon implementation. Sinceeach tether "node" has six degrees of freedom, and some complex simulations may require calculation ofelectrodynamic and aerodynamic forces at each segment, these small timesteps mean that an explicitpropagator can require a large amount of computational power, and detailed simulations may run veryslowly.To address this challenge, TetherSim also can utilize an implicit cable dynamics propagationalgorithm based upon finite element simulation methods. 1 This implicit method can use much largertimesteps and remain numerically stable.Because the temperature of the tether can fluctuate significantly due to solar heating and ohmicdissipation, the simulation uses a temperature-dependent model for the stress-strain behavior of thealuminum tether. The model also assumes that the tether has no torsional or flexural rigidity.Orbital Dynamics ModelThe code calculates the orbital motion of the satellite, endmass, and tether elements using a 4thorder Runge-Kutte algorithm to explicitly integrate the equations of motion according to Cowell’smethod. 2 The program uses an 8 th -order spherical harmonic model of the geopotential and a 1 st ordermodel for the lunar gravity. When a satellite enters the Moon’s sphere of influence, the trajectory isupdated using the lunar potential as the primary body and a 1 st order model of the geopotential as aperturbing force.lρ1. Hoyt, R.P., “A Stable Implicit Propagator for Space Tether Dynamics,” Appendix C in Stabilization of ElectrodynamicSpace Tethers, TUI final report on NASA Contract NAS8-01013.2. Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA, 1987, p. 447.M-2